Fig. 3(a) shows the temperature \(T\)  and the Mach number
on the equator of the steady flow in model Thick3. In this model, the gas is optically
thick throughout the flow except for the funnel region.
The luminosities in models Thick2 – Thick5 are chaotically variable with small amplitudes
of a few factor and the chaotic variation of the luminosity is attributed to the
Kelvin-Helmholtz instability which occurs along the centrifugal barrier surface.
The averaged luminosities are
as high as \(1.1 \times 10^{39}\)  – 2.7 \(\times 10^{40}\)  erg s\(^{-1}\) .
Such powerful radiation will influence the pre-shock and post-shock structures.
As far as 1D profiles of Fig. 3(a) are concerned,
the shock front seems to be discontinuous similarly to the adiabatic model AD, and
the large effects of the radiation on the shock structure seem not to be found there.
However, Fig. 3(b) shows an oblique shock wave with a finite thickness which
is elongated upward from \(r \sim 10\)  on the equator. The shock has a finite thickness of a few \(r_{g}\)  (several mesh points).
Fig. 4 shows 2D contours of the density (a) and the temperature (b) in model Thick5.
The overall flow consists of two regions; (1) the funnel region with an opening angle of \(\sim 15^\circ \)  between the rotational axis and the centrifugal barrier and (2) the gas accreting region between the centrifugal barrier and the equatorial plane.
The centrifugal barrier is roughly defined by a balance between the centrifugal force and the gravitational force.
The accreting region is separated into further two regions by the oblique shock wave.
The finite shock thickness is due to the radiative effect.
The figure shows the turbulent phenomana along the centrifugal barrier and the turbulent
gas becomes optically thick in the funnel region with increasing input density. The turbulent motion is originated
in the Kelvin-Helmholtz instability which can occur when there exists a velocity shear in a single continuous fluid or if there is a velocity
difference across the interface between two fluids [1]}. We confirm here the upward moving gas and the downward accreting gas across the centrifugal barrier above \(z \ge 10\)  in the funnel region.
In these models with high mass accretion rates and luminosities, the accretion disc is optically and geometrically thick
and it may be unfavourable for us to name the flow as “accretion disc" because \(h/r\)  \(\ge 1\) 
in the inner region of the flow, if we define the disc height to be a height at which the
density drops to one-tenth of the central density on the equatorial plane.
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